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Whitney Extension Theorem Smoothness and Smooth Extensions (I): Generalization of MWK Functions and Gradually Varied Functions Li Chen Department of Computer Science and Information Technology University of the District of Columbia [email protected] May 19, 2010 Abstract A mathematical smooth function means that the function has continuous derivatives to a certain degree C(k). We call it a k-smooth function or a smooth function if k can grow infinitively. Based on quantum physics, there is no such smooth surface in the real world on a very small scale. However, we do have a concept of smooth surfaces in practice since we always compare whether one surface is smoother than another one. This paper deals with the possible definitions of “natural” smoothness and their relationship to the original mathematical definition of smooth functions. The motivation of giving the definition of a smooth function is to study smooth extensions for practical applications. We observe this problem from two directions: From discrete to continuous, we suggest considering both micro smooth, the refinement of a smoothed function, and macro smooth, the best approximation using existing discrete space. (For two-dimensional or higher dimensional cases, we can use Hessian matrices.) From continuous to discrete, we suggest a new definition of natural smooth, it uses a scan from down scaling to up scaling to obtain the a ratio for sign changes by ignoring zero to represent the smoothness. For differentiable functions, mathematical smoothness does not mean a “good looking” smooth for a sampled set in discrete space. Finally, we discuss the Lipschitz continuity for defining the smoothness, which will be called discrete smoothness. This paper gives philosophical consideration of smoothness for practical problems, rather than a mathematical deduction or reduction, even though our inferences are based on solid mathematics. 1. Introduction The world can be described by continuity and discontinuity. Most parts are continuous. Discontinuity provides the boundary of continuous components, and continuity gives the content of discontinuous. An image can be separated as several continuous components. The edge of a component indicates the discontinuity to two parties. However, the boundary itself could be continuous. Continuity sometimes means controllable and discontinuity means uncontrollable such as something suddenly happened. For instance, a sunny day suddenly had a rain. Continuous sometime is not good enough to describe a phenomenal. Therefore, in continuous, we need to define smoothness. As well as in discontinuous, we want to define chaos. This paper only focuses on the concept of smooth and its extensions. Mathematicians always try to make things simpler. What is the smoothness? The mathematical smooth function means that a function has continuous derivatives to a certain degree C(k). We call it as a k-smooth function or the smooth function if k could be infinitively growth. Based on quantum physics, there is no such smooth surface in real world in quantum-scale. So the “natural” smoothness is relative not absolute. 1 On the other hand, a mathematical smooth function does not always mean the continuity in real world. For instance, when a sunny day starts to rain. The rain usually does not come by a centimeter in a minute. The rain usually had few drops then become bigger. So in a microscope, the weather changes continuously or smoothly per say. But the truth is that the sunny day had a rain! That is discontinued. It comes a problem, what shall we define the smoothness that can be used to real world problems and it does not course a contradiction to our beautiful mathematics exist. In fact, mathematician has already observed the problem. They have used Lipschitz conditions [15] and rectifiability to restrict a function to have a better continuous and smooth [11]. It was perfect amazing that three papers published in the same year dealing with a same or very similar problem. McShane, Whitney, and Kirszbraun all did independently in 1934. They obtained an important theorem for the Lipschitz function extension. What they proved is: For any subset E of a metric space S, if there is function f on E satisfies the Lipschitz condition, then it can be extended to S preserving the same Lipschitz constant. So our problem is solved! Actually it is not. McShane-Whitney-Kirszbraun Theorem only provides a theoretical result. This is because that the Lipschitz constant could be very big. No one who is involved in real problems wants to admit such a function is a “continuous” function. In 1983, Krantz published a paper that specifically studied smoothness of functions in Lipschitz space for technical applications [14]. Unfortunately, such an important movement did not get enough attentions in mathematics. Along with the fast development of computer science, scientists started to study “continuous functions” in a digital format for image processing. Rosenfeld pioneered this direction. In1982, he used a fuzzy logic concept to study the relationship among pixels. He defined the concept of fuzzy connected components. In 1985, Chen proposed another type of connectedness between two pixels called lambda-connectedness. This concept was found later having closer connections to continuous functions. In 1986, Rosenfeld proposed discretely continuous functions [16]. In 1989, Chen simplified lambda-connectedness to gradual variation and proved the necessary and sufficient condition of the extension [6]. In 1989, Steele used the Lipschitz condition to check the image “smoothness.” Chen’s constructive method suggests an algorithm that is totally different from McShane’s construction that builds a function from a maximum or minimum possibilities. The algorithm designed by Chen uses the local and middle construction that is more suitable to the practice. In addition, Chen used a sequence of real or rational numbers instead of integers, treating the problem more generally. That overcomes the limitation of Lipschitz space by enlarging to local Lipschitz [1-8]. What is our point? Even though we have a continuous or mathematical smooth function to describe the addressed question, the problem may only be able to be presented in a finite form for large amount of applications using finite samples in today’s technology. Based on McShane-Whitney-Kirszbraun theorem, a continuous surface always exists. According to the classic the Weierstrass Approximation Theorem: If f is a continuous real-valued function on [a,b], a closed and bounded interval, f can be uniformly approximated on that interval by polynomials to any degree of accuracy. In addition to the results obtained by Fefferman [12], 2 differentiable extensions of the Whitney’s problem are exists. However, the solutions are based on infinitively refinement of the domain space. In practice, we have the limited availability for the space we are dealing with. Recall the example of the sunny day had a rain, even though we could insert the time to micro second, we still have a discontinuous event that the sunny day usually does not have a rain. The mathematical definition of smooth functions does not provide a precise description of such phenomenal of real problems. The existence of a smooth function for “sudden rain” does not good enough to the problem of our sampling scale by minutes, or second. Smoothness is depending on sampling in real world or nature! For many cases absolute smoothness based on the mathematical definition may not be working right. This paper deals with the possible definitions of natural smoothness and their relationship with the original mathematical definition of smooth functions. The motivation of giving the definitions of smooth function is to study smooth extension for practical applications. We observe this problem in two directions: From discrete to continuous, we suggest to consider both of micro smooth meaning the refinement to a smoothed function, and macro smooth meaning the best approximation using existing discrete space. For two-dimensional or high dimensional cases, we can use the Hessian matrix to determine the extreme points and to represent the case. From continuous to discrete, we suggest a new definition of natural smooth, it uses a scan from down scaling to up scaling to get the a ratio for sign changes with ignoring zero to represent the smoothness. For differentiable functions, mathematical smoothness does not mean a “good looking” smooth for a sampled set in discrete space. Finally, we discuss the Lipschitz continuity for defining the smoothness. This paper is to give philosophical consideration of smoothness for practical problem, rather a mathematical deduction or reduction even though our inferences are based on solid mathematics. Since 2005, Fefferman started to publish papers for Whitney’s smooth extensions and suggest the technology to solve the real problems of finite sample points. This inspirited our recent research. Our resent research results suggest that we need to go one more step to clarify what real smooth function is in real world. 2. Review of Concepts We have reviewed many existing methods for continuous functions’ extension [2]. Despite many important practical methods, we first introduce here the key theorem by McShane, Whitney, and Kirszbraun. It gives a construction to a continuous extension to a set including a finite set. McShane gave a constructive proof for the existence of the extension in [15]. He constructed a minimal extension (INF) that is Lipschitz. Theorem 2.1 (The McShane-Whitney-Kirszbraun Theorem) Let J be a closed subset of D, and f(J) be a function. If | f (x) − f (y) |≤ Lip | x − y | where Lip is a constant for all x, y in J, then F(x) = maxa∈J { f (a) − Lip| a − x |} (2.1) is a continuous extension such that | F(x) − F(y) |≤ Lip | x − y | for all x, y in D. 3 We can also construct a maximum extension by F(x) = mina∈J { f (a) + Lip| a − x |} (2.2) We can call it a maximum extension (SUP).
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